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Theorem ssfin4 8182
Description: Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ssfin4  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  B  e. FinIV )

Proof of Theorem ssfin4
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  A  e. FinIV )
2 pssss 3434 . . . . . . . . 9  |-  ( x 
C.  B  ->  x  C_  B )
3 simpr 448 . . . . . . . . 9  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  B  C_  A
)
42, 3sylan9ssr 3354 . . . . . . . 8  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  ->  x  C_  A )
5 difssd 3467 . . . . . . . 8  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  ->  ( A  \  B )  C_  A )
64, 5unssd 3515 . . . . . . 7  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  ->  (
x  u.  ( A 
\  B ) ) 
C_  A )
7 pssnel 3685 . . . . . . . . 9  |-  ( x 
C.  B  ->  E. c
( c  e.  B  /\  -.  c  e.  x
) )
87adantl 453 . . . . . . . 8  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  ->  E. c
( c  e.  B  /\  -.  c  e.  x
) )
9 simpllr 736 . . . . . . . . . . 11  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  ->  B  C_  A )
10 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  -> 
c  e.  B )
119, 10sseldd 3341 . . . . . . . . . 10  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  -> 
c  e.  A )
12 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  ->  -.  c  e.  x
)
13 elndif 3463 . . . . . . . . . . . 12  |-  ( c  e.  B  ->  -.  c  e.  ( A  \  B ) )
1413ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  ->  -.  c  e.  ( A  \  B ) )
15 ioran 477 . . . . . . . . . . . 12  |-  ( -.  ( c  e.  x  \/  c  e.  ( A  \  B ) )  <-> 
( -.  c  e.  x  /\  -.  c  e.  ( A  \  B
) ) )
16 elun 3480 . . . . . . . . . . . 12  |-  ( c  e.  ( x  u.  ( A  \  B
) )  <->  ( c  e.  x  \/  c  e.  ( A  \  B
) ) )
1715, 16xchnxbir 301 . . . . . . . . . . 11  |-  ( -.  c  e.  ( x  u.  ( A  \  B ) )  <->  ( -.  c  e.  x  /\  -.  c  e.  ( A  \  B ) ) )
1812, 14, 17sylanbrc 646 . . . . . . . . . 10  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  ->  -.  c  e.  (
x  u.  ( A 
\  B ) ) )
19 nelneq2 2534 . . . . . . . . . 10  |-  ( ( c  e.  A  /\  -.  c  e.  (
x  u.  ( A 
\  B ) ) )  ->  -.  A  =  ( x  u.  ( A  \  B
) ) )
2011, 18, 19syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  ->  -.  A  =  (
x  u.  ( A 
\  B ) ) )
21 eqcom 2437 . . . . . . . . 9  |-  ( A  =  ( x  u.  ( A  \  B
) )  <->  ( x  u.  ( A  \  B
) )  =  A )
2220, 21sylnib 296 . . . . . . . 8  |-  ( ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  /\  ( c  e.  B  /\  -.  c  e.  x ) )  ->  -.  ( x  u.  ( A  \  B ) )  =  A )
238, 22exlimddv 1648 . . . . . . 7  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  ->  -.  ( x  u.  ( A  \  B ) )  =  A )
24 dfpss2 3424 . . . . . . 7  |-  ( ( x  u.  ( A 
\  B ) ) 
C.  A  <->  ( (
x  u.  ( A 
\  B ) ) 
C_  A  /\  -.  ( x  u.  ( A  \  B ) )  =  A ) )
256, 23, 24sylanbrc 646 . . . . . 6  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  x  C.  B )  ->  (
x  u.  ( A 
\  B ) ) 
C.  A )
2625adantrr 698 . . . . 5  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( x  u.  ( A  \  B
) )  C.  A
)
27 simprr 734 . . . . . . 7  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  x  ~~  B
)
28 difexg 4343 . . . . . . . 8  |-  ( A  e. FinIV  ->  ( A  \  B )  e.  _V )
29 enrefg 7131 . . . . . . . 8  |-  ( ( A  \  B )  e.  _V  ->  ( A  \  B )  ~~  ( A  \  B ) )
301, 28, 293syl 19 . . . . . . 7  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( A  \  B )  ~~  ( A  \  B ) )
312ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  x  C_  B
)
32 ssinss1 3561 . . . . . . . . . 10  |-  ( x 
C_  B  ->  (
x  i^i  A )  C_  B )
3331, 32syl 16 . . . . . . . . 9  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( x  i^i 
A )  C_  B
)
34 inssdif0 3687 . . . . . . . . 9  |-  ( ( x  i^i  A ) 
C_  B  <->  ( x  i^i  ( A  \  B
) )  =  (/) )
3533, 34sylib 189 . . . . . . . 8  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( x  i^i  ( A  \  B
) )  =  (/) )
36 disjdif 3692 . . . . . . . 8  |-  ( B  i^i  ( A  \  B ) )  =  (/)
3735, 36jctir 525 . . . . . . 7  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( ( x  i^i  ( A  \  B ) )  =  (/)  /\  ( B  i^i  ( A  \  B ) )  =  (/) ) )
38 unen 7181 . . . . . . 7  |-  ( ( ( x  ~~  B  /\  ( A  \  B
)  ~~  ( A  \  B ) )  /\  ( ( x  i^i  ( A  \  B
) )  =  (/)  /\  ( B  i^i  ( A  \  B ) )  =  (/) ) )  -> 
( x  u.  ( A  \  B ) ) 
~~  ( B  u.  ( A  \  B ) ) )
3927, 30, 37, 38syl21anc 1183 . . . . . 6  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( x  u.  ( A  \  B
) )  ~~  ( B  u.  ( A  \  B ) ) )
40 simplr 732 . . . . . . 7  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  B  C_  A
)
41 undif 3700 . . . . . . 7  |-  ( B 
C_  A  <->  ( B  u.  ( A  \  B
) )  =  A )
4240, 41sylib 189 . . . . . 6  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( B  u.  ( A  \  B ) )  =  A )
4339, 42breqtrd 4228 . . . . 5  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  ( x  u.  ( A  \  B
) )  ~~  A
)
44 fin4i 8170 . . . . 5  |-  ( ( ( x  u.  ( A  \  B ) ) 
C.  A  /\  (
x  u.  ( A 
\  B ) ) 
~~  A )  ->  -.  A  e. FinIV )
4526, 43, 44syl2anc 643 . . . 4  |-  ( ( ( A  e. FinIV  /\  B  C_  A )  /\  (
x  C.  B  /\  x  ~~  B ) )  ->  -.  A  e. FinIV )
461, 45pm2.65da 560 . . 3  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  -.  (
x  C.  B  /\  x  ~~  B ) )
4746nexdv 1941 . 2  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  -.  E. x
( x  C.  B  /\  x  ~~  B ) )
48 ssexg 4341 . . . 4  |-  ( ( B  C_  A  /\  A  e. FinIV )  ->  B  e.  _V )
4948ancoms 440 . . 3  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  B  e.  _V )
50 isfin4 8169 . . 3  |-  ( B  e.  _V  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
5149, 50syl 16 . 2  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  ( B  e. FinIV  <->  -. 
E. x ( x 
C.  B  /\  x  ~~  B ) ) )
5247, 51mpbird 224 1  |-  ( ( A  e. FinIV  /\  B  C_  A
)  ->  B  e. FinIV )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312    C. wpss 3313   (/)c0 3620   class class class wbr 4204    ~~ cen 7098  FinIVcfin4 8152
This theorem is referenced by:  domfin4  8183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-en 7102  df-fin4 8159
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